Strong Koszulness of Toric Rings Associated with Stable Set Polytopes

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(1) k[QG] is strongly Koszul. (2) G is a trivially perfect graph.

In particular, if k[QG] is strongly Koszul, then k[QG] is trivial. Throughout this note, we will use the standard terminologies of graph theory in [Diest]. 2. Strongly Koszul algebra

Let k be a field, R be a graded k-algebra, and m = R+ be the homogeneous maximal ideal of R. Definition 2.1 ([HeHiR]). A graded k-algebra R is said to be strongly Koszul if m admits a minimal system of generators {u1,...,ut} which satisfies the following condition:

For all subsequences ui1 ,...,uir of {u1,...,ut} (i1 ≤ ···≤ ir) and

for all j = 1,...,r − 1, (ui1 ,...,uij−1 ): uij is generated by a subset of elements of {u1,...,ut}. A graded k-algebra R is called Koszul if k = R/m has a linear resolution. By the following theorem, we can see that a strongly Koszul algebra is Koszul. Proposition 2.2 ([HeHiR, Theorem 1.2]). If R is strongly Koszul with respect to the minimal homogeneous generators {u1,...,ut} of m = R+, then for all subsequences

{ui1 ,...,uir } of {u1,...,ut}, R/(ui1 ,...,uir ) has a linear resolution. The following proposition plays an important role in the proof of the main theorem. Theorem 2.3 ([HeHiR, Proposition 2.1]). Let S be a semigroup and R = k[S] be the semigroup ring generated by S. Let {u1,...,ut} be the generators of m = R+ which correspond to the generators of S. Then, if R is strongly Koszul, then for all subsequences {ui1 ,...,uir } of {u1,...,ut}, R/(ui1 ,...,uir ) is also strongly Koszul. By this theorem, we have

Corollary 2.4. If k[QG] is strongly Koszul, then k[QGW ] is strongly Koszul for all induced subgraphs GW of G. 2 3. Hibi ring and comparability graph In this section, we introduce the concepts of a Hibi ring and a comparability graph. Both are defined with respect to a partially ordered set. Let P = {p1,...,pn} be a finite partially ordered set consisting of n elements, which is referred to as a poset. Let J(P ) be the set of all poset ideals of P , where a poset ideal of P is a subset I of P such that if x ∈ I, y ∈ P , and y ≤ x, then y ∈ I. Note that ∅ ∈ J(P ). First, we give the definition of the Hibi ring introduced by Hibi.

Deﬁnition 3.1 ([Hib]). For a poset P = {p1,...,pn}, the Hibi ring Rk[P ] is deﬁned as follows:

Rk[P ] := k[T · Y Xi | I ∈ J(P )] ⊂ k[T,X1,...,Xn] i∈I Example 3.2. Consider the following poset P = (1 ≤ 3, 2 ≤ 3 and 2 ≤ 4).

{1, 2, 3, 4} •❏ tt ❏❏❏ tt ❏❏ ttt ❏❏ •❏ • •t❏t ❏•❏ 3 ❏❏ 4 {1, 2, 3} ❏❏ tt ❏❏{1, 2, 4} ❏❏ ❏❏ tt ❏❏ ❏❏ ❏❏ tt ❏❏ ❏❏ ❏❏ tt ❏❏ P = ❏❏ J(P )= {1, 2} t•t❏❏ t• {2, 4} ❏❏ tt ❏❏❏ tt ❏❏ tt ❏❏ tt ❏❏ ttt ❏❏ ttt 1 • • 2 {1} •❏t❏ •t {2} ❏❏ tt ❏❏ ttt ❏❏ tt ❏•tt ∅ Then we have

Rk[P ]= k[T,TX1,TX2,TX1X2,TX2X4,TX1X2X3,TX1X2X4,TX1X2X3X4]. Hibi showed that a Hibi ring is always normal. Moreover, a Hibi ring can be represented as a factor ring of a polynomial ring: if we let

IP := (XI XJ − XI∩J XI∪J | I,J ∈ J(P ),I 6⊆ J and J 6⊆ I) be the binomial ideal in the polynomial ring k[XI | I ∈ J(P )] defined by a poset P , then Rk[P ] =∼ k[XI | I ∈ J(P )]/IP . Hibi also showed that IP has a quadratic Gröbner basis for any term order which satisfies the following condition: the initial term of XI XJ −XI∩J XI∪J is XI XJ . Hence a Hibi ring is always Koszul from general theory. Next, we introduce the concept of a comparability graph. Definition 3.3. A graph G is called a comparability graph if there exists a poset P which satisfies the following condition: {i, j} ∈ E(G) ⇐⇒ i ≥ j or i ≤ j in P. We denote the comparability graph of P by G(P ). 3 Example 3.4. The lower-left poset P defines the comparability graph G(P ).

•❄ • •❇✬ ❄❄ ⑧⑧ ⑤⑤✗✬❇❇ ❄❄ ⑧⑧ ⑤⑤ ✗ ✬ ❇❇ ❄❄ ⑧⑧ ⑤⑤ ✗✗ ✬✬ ❇❇ ❄❄ ⑧⑧ ⑤⑤ ✗ ✬ ❇❇ P = •⑧❄ G(P )= •⑤❑❑ ✗ ✬ s • ⑧⑧ ❄❄ ❑❑ ✗ ✬ ss ⑧⑧ ❄❄ ❑❑✗ ✬ss ⑧ ❄❄ ✗ ❑❑ss✬ ⑧⑧ ❄ ✗✗ ss❑❑✬✬ •⑧⑧ ❄• •s✗ s ❑•

′ ′ Remark 3.5. It is possible that P =6 P but G(P ) = G(P ). Indeed, for the ′ ′ following poset P , G(P ) is identical to G(P ) in the above example.

•❄ ⑧⑧ ❄❄ ⑧⑧ ❄❄ ⑧⑧ ❄❄ ′ ⑧⑧ ❄❄ P = •❖⑧❖ ♦• ❖❖❖ ♦♦♦ ❖❖♦❖♦♦ ♦♦♦ ❖❖❖ •♦♦♦ ❖❖•

Complete graphs are comparability graphs of totally ordered sets. Bipartite graphs and trivially perfect graphs (see the next section) are also comparability graphs. Moreover, if G is a comparability graph, then the suspension (e.g., see [HiNOS, p.4]) of G is also a comparability graph. Recall the following deﬁnitions of two types of polytope which are deﬁned by a poset.

Deﬁnition 3.6 (see [St1]). Let P = {p1,...,pn} be a ﬁnite poset. (1) The order polytope O(P ) of P is the convex polytope which consists of n (a1,...,an) ∈ R such that 0 ≤ ai ≤ 1 with ai ≥ aj if pi ≤ pj in P . (2) The chain polytope C(P ) of P is the convex polytope which consists of Rn (a1,...,an) ∈ such that 0 ≤ ai ≤ 1 with ai1 + ··· + aik ≤ 1 for all

maximal chain pi1 < ···

{The vertices of O(P )} = {ρ(I) | I is a poset ideal of P }, {The vertices of C(P )} = {ρ(A) | A is an anti-chain of P },

where A = {pi1 ,...,pik } is an anti-chain of P if pis 6≤ pit and pis 6≥ pit for all s =6 t. Hence we have QG(P ) = C(P ). In [HiL], Hibi and Li answered the question of when C(P ) and O(P ) are unimod- ularly equivalent. From their study, we have the following theorem. Theorem 3.7 ([HiL, Theorem 2.1]). Let P be a poset and G(P ) be the comparability graph of P . Then the following are equivalent: (1) The X-poset in Example 3.4 does not appear as a subposet (refer to [St2, Chapter 3]) of P . ∼ (2) Rk[P ] = k[QG(P )]. 4 Example 3.8. The cycle of length 4 C4 and the path of length 3 P4 are comparability graphs of Q1 and Q2, respectively.

•❄❄ ⑧• •❄❄ • ❄❄ ⑧⑧ ❄❄ ❄❄ ⑧⑧ ❄❄ ❄❄ ⑧⑧ ❄❄ Q1 = ⑧❄❄⑧ Q2 = ❄❄ ⑧⑧ ❄❄ ❄❄ ⑧⑧ ❄❄ ❄❄ ⑧⑧ ❄❄ ❄❄ •⑧⑧ ❄ • •❄ •

∼ ∼ Hence k[QC4 ] = Rk[Q1] and k[QP4 ] = Rk[Q2]. A ring R is trivial if R can be constructed by starting from polynomial rings and repeatedly applying tensor and Segre products. Herzog, Hibi and Restuccia gave an answer for the question of when is a Hibi ring strongly Koszul.

Theorem 3.9 (see [HeHiR, Theorem 3.2]). Let P be a poset and R = Rk[P ] be the Hibi ring constructed from P . Then the following assertions are equivalent: (1) R is strongly Koszul. (2) R is trivial. (3) The N-poset as described below does not appear as a subposet of P .

•❖❖❖ • ❖❖❖

❖ ❖❖❖ • ❖❖•

By this theorem, Corollary 2.4, and Example 3.8, we have

Corollary 3.10. If G contains C4 or P4 as an induced subgraph, then k[QG] is not strongly Koszul.

4. trivially perfect graph In this section, we introduce the concept of a trivially perfect graph. As its name suggests, a trivially perfect graph is a kind of perfect graph; it is also a kind of comparability graph, as described below. Deﬁnition 4.1. For a graph G, we set α(G) := max{#S | S is a stable set of G}, m(G) := #{the set of maximal cliques of G}.

We call α(G) the stability number (or independence number) of G. In general, α(G) ≤ m(G). Moreover, if G is chordal, then m(G) ≤ n by Dirac’s theorem [Dir]. In [G], Golumbic introduced the concept of a trivially perfect graph.

Definition 4.2 ([G]). We say that a graph G is trivially perfect if α(GW )= m(GW ) for any induced subgraph GW of G. 5 For example, complete graphs and star graphs (i.e., the complete bipartite graph K1,r) are trivially perfect. We define some additional concepts related to perfect graphs. Let CG be the set of all cliques of G. Then we define

ω(G) := max{#C | C ∈ CG},

θ(G) := min{s | C1 a ··· a Cs = V (G),Ci ∈ CG}, χ(G) := θ(G), where G is the complement of G. These invariants are called the clique number, clique covering number, and chromatic number of G, respectively. In general, α(G) = ω(G), θ(G) ≤ m(G) and ω(G) ≤ χ(G). The deﬁnition of a perfect graph is as follows.

Deﬁnition 4.3. We say that a graph G is perfect if ω(GW )= χ(GW ) for any induced subgraph GW of G. Lov´asz proved that G is perfect if and only if G is perfect [Lo]. The theorem is now called the weak perfect graph theorem. With it, it is easy to show that a trivially perfect graph is perfect. Proposition 4.4. Trivially perfect graphs are perfect. Proof. Assume that G is trivially perfect. By [Lo], it is enough to show that G is perfect. For all induced subgraphs GW of G, we have

m(GW )= α(GW )= ω(GW ) ≤ χ(GW )= θ(GW ) ≤ m(GW ) by general theory (note that GW = GW ). Golumbic gave a characterization of trivially perfect graphs. Theorem 4.5 ([G, Theorem 2]). The following assertions are equivalent: (1) G is trivially perfect. (2) G is C4,P4-free, that is, G contains neither C4 nor P4 as an induced subgraph.

Proof. (1) ⇒ (2): It is clear since α(C4) = 2, m(C4) = 4, and α(P4) = 2, m(P4) = 3. (2) ⇒ (1): Assume that G contains neither C4 nor P4 as an induced subgraph. If G is not trivially perfect, then there exists an induced subgraph GW of G such that α(GW ) < m(GW ). For this GW , there exists a maximal stable set SW of GW which satisﬁes the following:

There exists s ∈ SW such that s ∈ C1 ∩ C2 for some distinct pair of cliques C1,

C2 ∈ CGW .

Note that #SW > 1 since GW is not complete. Then there exist x ∈ C1 and y ∈ C2 such that {x, s}, {y,s} ∈ E(GW ) and {x, y} 6∈ E(GW ). Let u ∈ SW \{s}. If {x, u} ∈ E(GW ) or {y,u} ∈ E(GW ), then the induced graph G{x,y,s,u} is C4 or P4, a contradiction. Hence {x, u} 6∈ E(GW ) and {y,u} 6∈ E(GW ). Then {x, y}∪{S \{s}} is a stable set of GW , which contradicts that S is maximal. Therefore, G is trivially perfect. 6 Next, we show that a trivially perfect graph is a kind of comparability graph. First, we define the notion of a tree poset. Definition 4.6 (see [W]). A poset P is a tree if it satisfies the following conditions: (1) Each of the connected components of P has a minimal element. ′ (2) For all p, p ∈ P , the following assertion holds: if there exists q ∈ P such ′ ′ ′ that p,p ≤ q, then p ≤ p or p ≥ p . Example 4.7. The following poset is a tree:

• • • tt tt tt tt tt tt tt tt tt tt •❏❏ •t t•t ❏❏ tt ❏❏ tt ❏❏ tt ❏❏ tt ❏•tt

Tree posets can be characterized as follows. Proposition 4.8. Let P be a poset. Then the following assertions are equivalent: (1) P is a tree. (2) Neither the X-poset in Example 3.4, the N-poset in Theorem 3.9, nor the diamond poset as described below appears as a subposet of P .

•❏ ttt ❏❏ ❏ •t❏tt ❏❏• ❏❏ ttt ❏ ❏❏•ttt

In [W], Wolk discussed the properties of the comparability graphs of a tree poset and showed that such graphs are exactly the graphs that satisfy the “diagonal condition”. This condition is equivalent to being C4, P4-free, and hence we have Corollary 4.9. Let G be a graph. Then the following assertions are equivalent: (1) G is trivially perfect. (2) G is a comparability graph of a tree poset. (3) G is C4, P4-free. Remark 4.10. A graph G is a threshold graph if it can be constructed from a one-vertex graph by repeated applications of the following two operations: (1) Add a single isolated vertex to the graph. (2) Take a suspension of the graph. The concept of a threshold graph was introduced by Chv´atal and Hammer [CHam]. They proved that G is a threshold graph if and only if G is C4, P4, 2K2-free. Hence a trivially perfect graph is also called a quasi-threshold graph.

7 5. Proof of Main theorem In this section, we prove the main theorem. Theorem 5.1. Let G be a graph. Then the following assertions are equivalent:

(1) k[QG] is strongly Koszul. (2) G is trivially perfect. Proof. We assume that G is trivially perfect. Then there exists a tree poset P such that G = G(P ) from Corollary 4.9. This implies that neither the X-poset in Example 3.4 nor the N-poset in Theorem 3.9 appears as a subposet of P by Proposition 4.8 ∼ , and hence k[QG(P )] = Rk[P ] is strongly Koszul by Theorems 3.7 and 3.9. Conversely, if G is not trivially perfect, G contains C4 or P4 as an induced subgraph by Corollary 4.9. Therefore, we have that k[QG] is not strongly Koszul by Corollary 3.10.

6. Remark on usual Koszulness of k[QG]

It seems to be diﬃcult to give a complete characterization of when k[QG] is Koszul. However, it is known that k[QG] is Koszul for many graphs G.

Theorem 6.1 ([EN]). If G is an almost bipartite graph, then k[QG] is Koszul, where a graph G is almost bipartite if there exists a vertex v ∈ [n] such that the induced subgraph G[n]\v is bipartite, that is, does not contain induced odd cycles. Remark 6.2. An almost bipartite graph is one such that all its odd cycles share a common vertex. Hence if G is almost bipartite, then G is K4-free, that is, ω(G) ≤ 3. In the case of n ≤ 5, G is almost bipartite if and only if G is K4-free. Next, we recall the theorem of Hibi and Li (Theorem 3.7). A graph G is an HL-comparability graph if it is the comparability graph of a poset P which does not contain the X-poset in Example 3.4 as a subposet. From their theorem, we have

Theorem 6.3. If G is an HL-comparability graph, then k[QG] is Koszul. Remark 6.4. (1) If n ≤ 5, the notion of HL-comparability is equivalent to the usual comparability. (2) Bipartite graphs are comparability graphs defined by posets with rank P ≤ 1. Hence bipartite graphs are HL-comparability graphs. r (3) Let G be a complete r-partite graph with V (G) = Qi=1 Vi. Then G is an HL-comparability graph if and only if #{Vi | #Vi =1} ≥ r − 2. (4) Let G be a closed graph (see [HeHiHrKR]) which satisfies the following condition: for all C1,C2 ∈ CG, #{C1 ∩C2} ≤ 1. Then G is an HL-comparability graph. As the end of this note, we give a classification table of connected six-vertex graphs using Harary [Har].

8 Classiﬁcation - six-vertex (112 items) Almost Bipartite • •✴✴ •❏ •✴❏ ✴ •✴✴ •✴✴ • •✴✴ •❏t•✴❏ ✴ •❏❏ •✴✴ ✎✎ ✎✎ ✴ ✎✎ ❏❏❏✴ ✎✎ ✴ ✴ ✎✎ ✴ ✎t✎tt❏❏❏✴ ✎✎ ❏✎✎❏❏✴ •✎ ✎✎ • •✎✴✴ ✎• •✎✴✴ ✴✴ • •✎✴✴ ✎• •t✎✴✴ ✎• •✎✴✴ ✎✎ ✎• •✎✎ • ✴• •✎✎ ✴• ✴• ✴• •✎✎ ✴• •✎✎ ✴•✎✎ •✎✎ Comparability HL-Comparability •✴ •✴ • • ✎ ✴ ✴✴ ✎ttt✎ Bipartite •✎✴✎ ✴✴ • •✎✴❏t✎t ✎✎ • ✴ ✴ ✎ ✴❏❏✎❏ ✎ ✴• ✴•✎✎ ✴•✎✎ ❏•✎✎ •❏❏ • •❏❏ • •✴ t• •✴ •✴ •✴ •✴ ❏❏ ✎✎ ❏❏ ✎✎t✴✴t ✎✎ ✴✴ ✎ ✴✴ ✎✎ ✴✴ ✎ ✴✴ • •✴ •✴ •✴ •✴ ❏• •✎ ❏• •t✎✴t ✴ • •✎✴ ✴✎✎ • •✎✴ ✎✴✎ • ✎ ✴✴ ✎t✴✴tt✴✴ ✴✴ ✎ ✎ ✴✴ ✴✴ ✎ ✴✴ ✎✴✴ ✎ ✴✴ ✎✴✴ ✎ •✎✴✎ • •❏t✎✴✎t ✴ • • •✎✎ • •✎✎ • •✎✎ •✎✎ •✎✎ •✎✎ •✎✎ ✴ ✎ ✴❏❏❏✴ ✎ ✴• •✎✎ ✴• ❏✴•✎✎ •✴ • • •✴✴ •✴❏❏ • • •✴ • •✴ • •✴ ✎✎ ✴✴ ✎✎ ✴✴ ✴✴ ✎✎ ✴✴❏❏ ✴✴ ✎✎ ✴✴ ✎✎ ✎ ✴✴ Trivially •✴ • •❏❏ •✴ •✎✴ ✴ • •✎✴ ✴ • •✎✴ ✴ ❏• •✴ • •✎✴ • •✎✴ ✎✎ • ✴✴ ✎ ✎tt❏t❏✴✴ ✴✴ ✴✴ ✎ ✴✴ ✴✴ ✎ ✴✴ ✴✴ ✴✴ ✴✴ ✎ ✴✴ ✎ ✎ •✴ ✴✎✎ • •❏t✎✴✎t ❏• • •✎✎ • •✎✎ • • • • • •✎✎ •✎✎ •✎✎ Perfect ✴ ✎✴ ✎ ✴❏❏❏tt✎ ✴•✎✎ ✴•✎✎ ✴•ttt❏•✎✎ • • • • • •✴ • • • •✴ • t•✴ •✴ • •❏❏t•✴✴ ✎✎ ✎✎ ✎✎ ✴✴ ✎✎ ✎ ✎✎ ✴✴ tt✎ ✴✴ ✎✎ ✴✴ ✎✎t✴✴t❏❏✴✴ •✴ •✴ •❏❏ •✴ •✎✴ • •✎✴ • •✎ • •✎✴ ✎✎ • •✎ • •tt ✎✎ • •❏✎✴ ✴ • •❏t✎✴t ✴ ❏• ✎ ✴✴ ✴✴ ✎ ❏❏✴✴ ✴✴ ✎ ✴✴ ✎ ✴✴ ✎ttt ✎ ✴❏✴❏❏✴✴ ✎ ✴❏✴❏❏t✴✴tt✎ •❏✎✴✎ ✴ • •❏✎✴✎ ❏• • •✎✎ • •✎✎ • • •t✎✎t • • • •✎✎ • • ❏•✎✎ •tt❏•✎✎ ✴❏❏❏✴ ✴❏❏❏tt✎ ✴• ❏✴• ✴•ttt❏•✎✎ • •✴ •✴ •✴ •✴ •✴ • •✴ •❏❏ • • •✴ • •✴ •✴❏❏ • •✴ •✴ •✴❏❏ •✴ ✎✎ ✴✴ ✎✎ ✴✴ ✴✴ ✎ ✴✴ ✴✴ ✎ ✴✴ ✎ ❏❏ ✎ ✴✴ ✎ ✴✴ ✎ ✴✴❏❏ ✎ ✴✴ ✴✴ ✎ ✴✴❏❏✴✴ •✎✴ • •❏✎✴ ✴ • •✎✴✎ ✴ • •✎✴✎ • •✎✴✎ ❏• •✎✴✎ • •✎✴✎ • •✎✴✎ ✴ ❏• •❏✎✴✎ ✴ • •✎✴❏✎ ✴ ❏• ✴✴ ✎ ✴❏✴❏❏t✴✴tt✎ ✴ ✴ ✴ ✴ ✴ ✴ ✎ ✴ ✴ ✴❏❏❏✴ ✎ ✴❏❏❏✴tt✎ • •✎✎ •tt❏•✎✎ ✴• ✴• ✴• • ✴• • ✴• • ✴• •✎✎ ✴• ✴• ✴• ❏✴•✎✎ ✴•ttt❏✴•✎✎ •✴ • •✴❏❏t•✴ • • •✴ •✴ • • • •✴ •✴ •✴ • • •✴ • •✴❏❏ •✴ ✎✎ ✴✴ ✎ ✎✎t✴✴t❏❏✴✴ ✎ ✎ ✎ ✴✴ ✴✴ ✎ ✎ ✎ ✎ ✴✴ ✎ ✴✴ ✴✴ ✎ ✎ ✎ ✴✴ ✎ ✴✴❏✎❏✴✴ •✎✴ ✴✎✎ • •✎✴tt ✴ ❏• •✎✴✎ ✎✎ • •✎✴✎ ✴ • •✎✎ ✎✎ • •✎✎ ✎✎ • •✎✴✎ ✴ • •✎✴✎ ✎✎ • •❏✎✴✎ ✴ • •✎✴✎ ✎✴✎ ❏• ✴✴ ✎✴✴ ✎ ✴✴ t✴✴tt✎ ✴ ✎ ✎ ✴ ✴ ✎ ✎ ✎ ✴ ✴ ✴ ✎ tt✎ ✴❏❏❏✴ ✎ ✴ ✎✴tt✎ •✎✎ •✎✎ •tt •✎✎ ✴•✎✎ •✎✎ ✴• ✴• •✎✎ •✎✎ •✎✎ • ✴• ✴• ✴•✎t✎tt•✎✎ ✴• ❏✴•✎✎ ✴•t✎✎tt✴•✎✎ • •✴ •✴ •✴ • •✴ • •✴ • •✴✴ • • •✴ • •✴ • •❏ •✴❏ •✴ •✴ ✎✎ ✎ ✴✴ ✎✎ ✴✴ ✎ ✴✴ ✎ ✴✴ ✎ ✎ ✴✴ ✎ ✴✴ ✴✴ ✎ ✎ ✴✴ ✎ ✴✴ ✎tt❏t❏✴✴ ✎t✴✴tt✎ ✴✴ •✎✴ ✎✎ • •✎✴ ✎✴✎ • •✴ ✎✎ • •✎✴✎ ✎✎ • •✎✴✎ ✴ • •✎✴✎ • •✎✴✎ ✴ • •❏✎✴✎ ✴ • •✎✴❏t✎t ❏• •✎✴t❏✎t ✎✴✎ • ✴✴ ✎ ✎ ✴✴ ✎✴✴ ✎ ✴ ✎ ✴ ✎ ✴ ✴ ✎ ✴ ✎ ✴ ✴ ✎ ✴❏❏❏✴ ✎ ✴❏❏❏tt✎ ✴❏❏✎❏✴tt✎ •✎✎ •✎✎ •✎✎ •✎✎ ✴•✎✎ • ✴•✎✎ • ✴• ✴•✎✎ ✴• •✎✎ ✴• ✴•✎✎ ✴• ❏✴•✎✎ ✴•ttt❏•✎✎ ✴•t✎✎tt❏✴•✎✎ • t• •✴❏❏t•✴ •✴ •✴ •✴ • •✴ • •✴ • •✴ •✴ • •✴✴ •✴ • •✴❏❏ •✴ ✎✎tt✎ ✎✎t✴✴t❏❏✴✴ ✎ ✴✴ ✴✴ ✎ ✴✴ ✎ ✴✴ ✎ ✴✴ ✎ ✎ ✴✴ ✴✴ ✎ ✴✴ ✴✴ ✎ ✴✴ ✎ ✎t✴✴t❏t✎❏✴✴ •t✎✴t ✎✎ • •❏t✎✴t ✴ ❏• •✎✴✎ ✴ • •✎✴✎ ✴ • •❏✎✴✎ ✴ • •✎✴✎ ✴✎✎ • •✎✴✎ ✴ • •❏✎✴✎ ✴ • •❏✎✴✎ ✎✴✎ • •✎✴t❏✎t ✎✴✎ ❏• ✴✴ ✎ ✎ ✴❏✴❏❏t✴✴tt✎ ✴ ✴ ✴ ✴ ✎ ✴❏❏❏✴ ✎ ✴ ✎✴ ✎ ✴ ✴ ✎ ✴❏❏❏✴ ✎ ✴❏❏✎❏✴ ✎ ✴❏❏✎❏✴tt✎ •✎✎ •✎✎ •tt❏•✎✎ ✴• ✴• ✴• ✴•✎✎ ✴• ❏✴•✎✎ ✴•✎✎ ✴•✎✎ ✴• ✴•✎✎ ✴• ❏✴•✎✎ ✴•✎✎ ❏✴•✎✎ ✴•t✎✎tt❏✴•✎✎ • t•✴ •❏❏ •✴ • • •✴ •✴ •❏ •✴❏ •✴ • •❏ •✴❏ •✴ • •✴ • •✴❏❏ •✴ ✎✎tt ✴✴ ✎✎ ❏✎❏✴✴ ✎ ✎ ✴✴ ✎ ✴✴ ✎tt❏t❏✴✴ ✎ ✴✴ ✎ ✎tt❏t❏✴✴ ✎t✴✴tt ✎t✴✴tt✎ ✎t✴✴t❏t✎❏✴✴ •t✎✴t • •❏✎✴ ✎✎ ❏• •❏✎✴✎ • •❏✎✴✎ ✎✴✎ • •✎✴t✎t ❏• •✎✴✎ ✴✎✎ • •t✎✴✎t ❏• •❏t✎✴✎t ✴ • •❏t✎✴✎t ✎✴✎ • •✎✴t❏✎t ✎✴✎ ❏• ✴✴ ✎ ✴❏✴❏✎❏ttt✎ ✴❏❏❏ ✎ ✴❏❏✎❏✴ ✴ tt✎ ✴ ✎✴ ✎ ✴ ✎ ✴❏❏❏✴ ✎ ✴❏❏✎❏✴ ✎ ✴❏❏✎❏✴tt✎ • •✎✎ •t✎✎t❏•✎✎ ✴• ❏•✎✎ ✴•✎✎ ❏✴• ✴•ttt•✎✎ ✴•✎✎ ✴•✎✎ ✴• •✎✎ ✴• ❏✴•✎✎ ✴•✎✎ ❏✴•✎✎ ✴•t✎✎tt❏✴•✎✎ • t•✴ •✴❏❏t•✴ •❏❏ •✴ •✴❏❏ •✴ •❏❏ •✴ • •✴ •❏❏ •✴ •❏❏ •✴ •❏❏ •✴ ✎✎tt✎ ✴✴ ✎✎t✴✴t❏❏✴✴ ✎ ❏❏✴✴ ✎ ✴✴❏❏✴✴ ✎tt❏t❏✴✴ ✎ ✴✴ ✎tt❏t❏✴✴ ✎tt❏t❏✴✴ ✎tt❏t❏✴✴ •t✎✴t ✎✎ • •❏✎✴tt ✴ ❏• •✎✴✎ ❏• •✎✴✎ ✴ ❏• •✎✴t✎t ❏• •✎✴✎ • •❏t✎✴✎t ❏• •t✎✴❏✎t ❏• •❏✎✴t✎t ❏• ✴✴ ✎ ✎ ✴❏✴❏❏t✴✴tt✎ ✴ ✎ ✴ ✴ ✎ ✴ tt✎ ✴ tt✎ ✴❏❏❏tt✎ ✴❏❏❏tt✎ ✴❏❏❏tt✎ •✎✎ •✎✎ •tt❏•✎✎ ✴• •✎✎ ✴• ✴•✎✎ ✴•ttt•✎✎ ✴•ttt•✎✎ ✴•ttt❏•✎✎ ✴•ttt❏•✎✎ ✴•ttt❏•✎✎ •✴ •✴ •✴ •✴ • • •✴ •✴ •✴ • •✴ •✴ •✴ •✴ ✎✎ ✴✴ ✎ ✴✴ ✎✎ ✴✴ ✎ ✴✴ ✎ ✎ ✎ ✴✴ ✎ ✴✴ ✎ ✴✴ ✎ ✎ ✴✴ ✴✴ ✎ ✴✴ ✎ ✴✴ •✎✴ ✴✎✎ • •❏✎✴ ✴✎✎ • •❏✎✴✎ ✎✎ • •✎✴✎ ✴✎✎ • •❏✎✴✎ ✎✴✎ • •✎✴✎ ✴ • •❏✎✴✎ ✎✴✎ • ✴✴ ✎✴✴ ✎ ✴❏✴❏✎❏t✴✴tt✎ ✴❏❏✎❏ ✎ ✴ ✎✴ ✎ ✴❏❏✎❏✴ ✎ ✴ ✴ ✎ ✴❏❏✎❏✴ ✎ •✎✎ •✎✎ •t✎✎t❏•✎✎ ✴•✎✎ ❏•✎✎ ✴•✎✎ ✴•✎✎ ✴•✎✎ ❏✴•✎✎ ✴• ✴•✎✎ ✴•✎✎ ❏✴•✎✎ •✴ •✴ •✴❏❏t•✴ •✴ •✴ •✴ •✴ •✴ •✴ ✎✎ ✴✴ ✴✴ ✎✎t✴✴t❏✎❏✴✴ ✎t✴✴tt✴✴ ✎ ✴✴ ✎ ✴✴ ✎t✴✴tt✎ ✴✴ •❏✎✴ ✴ • •❏✎✴tt ✴✎✎ ❏• •t✎✴✎t ✴ • •✎✴✎ ✎✴✎ • •❏✎✴t✎t ✴✎✎ • ✴❏✴❏❏✴✴ ✎ ✴❏✴❏✎❏t✴✴tt✎ ✴ ✴ ✎ ✴ ✎✴ ✎ ✴❏❏✎❏✴ ✎ • ❏•✎✎ •t✎✎t❏•✎✎ ✴• ✴•✎✎ ✴•✎✎ ✴•✎✎ ✴•✎✎ ❏✴•✎✎

•❏ •✴ •❏ •✴ •❏ •✴ ✎ ❏t❏t❏✴✴ ✎ ❏t❏t✎❏✴✴ ✎ ❏t❏t❏✴✴ •✎✴❏t✎tt ❏• •❏t✎✴✎tt✎✎ ❏• •❏t✎✴✎tt ❏• ✴❏❏❏ ✎ ✴❏❏✎❏tt✎ ✴❏❏❏tt✎ ✴• ❏•✎✎ ✴•t✎✎tt❏•✎✎ ✴•ttt❏•✎✎

•❏ •✴ • •✴ • •✴ •✴ •✴ •✴ •✴ ✎ ❏t❏t❏✴ ✎ tt✴ ✎ ✴ ✎ ✴tt✴ ✎ ✴ ✴ •t✎✴✎tt ❏✴• •❏t✎✴✎tt ✴• •✎✴✎ ✴• •t✎✴✎tt✴✴ ✴• •❏✎✴✎ ✴✴ ✴• ✴ ✎ ✴❏❏❏ ✎ ✴ ✎ ✴ ✴ ✎ ✴❏❏❏✴tt✎ ✴• •✎✎ ✴• ❏•✎✎ ✴• •✎✎ ✴• ✴•✎✎ ✴•ttt❏✴•✎✎

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(Kazunori Matsuda) Department of Mathematics, College of Science, Rikkyo Uni- versity, Toshima-ku, Tokyo 171-8501, Japan E-mail address: [email protected]